Flight-Propulsion Integration Dynamic Analysis and Adaptive Control of The Hypersonic Vehicle at Wide-range Mach Numbers

There are serious among various components of the hypersonic vehicle at wide-range Mach numbers, and the coupling among aerodynamic shape, propulsion system, structure, and control system is remarkable. For excellent performance, its components and functions are highly integrated, which brings serious challenges to traditional design examples, methods, and tools. To solve the integrated control problem of flight-propulsion coupling, a controller based on pole assignment is designed in this paper, and the control parameters can be adjusted according to real-time aerodynamic data. The parametric model of hypersonic vehicle at wide-range Mach numbers with coupling characteristics is established and the flight data under various working conditions are obtained. The flight dynamical model is established with a coupling relationship between the airframe and engine. Combined with the characteristic parameters of the aircraft attitude control closed-loop system, the control gain is designed based on expected poles and adjusted in real-time. The comparative simulations are carried out, and the results show that the control gain adjustment strategy based on aerodynamic parameter identification can improve the attitude angle command tracking quality and the robustness of the system.


I. INTRODUCTION
The hypersonic vehicle at wide-range Mach numbers refers to an aircraft whose flying Mach number is greater than 5, using a scramjet engine and its combined power, capable of maneuvering in the adjacent space at a height of 20-100km from sea level and relying on a rare atmosphere to perform missions such as strikes, investigations, carrying and confrontation [1][2][3][4]. With characteristics of high flight speed, rapid response, wide speed range, large lift-drag ratio, and big specific impulse, it shows remarkable advantages compared with conventional aircraft and has important military and civil value.
The scramjet engine has the characteristics of multivariable, non-linear, and time-varying which takes the difficulty to its control system. The coupling and mutual influence of aerodynamic force, engine working state of the hypersonic vehicle at wide-range Mach numbers are mainly expressed in that the flight attitude affects the thrust by the air mass flow rate of the inlet [5][6][7]. Meanwhile, thrust affects hypersonic vehicle attitude in two aspects. First, due to the special structure of the combined power engine, the thrust center can't be set at the centroid position of aircraft, which brings about an eccentric moment. Second, the highspeed expansion airflow from the nozzle acts on the asymmetric outer nozzle, resulting in an additional moment. Therefore, the coordinated control of attitude, speed, and thrust is particularly important [8][9][10]. Because of the particularity of the wide-range velocity vehicle, the conventional strategy usually used for the control system is not suitable for the application [11]. During the flight of an aircraft in a large altitude and speed range, the aerodynamic and thrust coefficients will change nonlinearly, which also brings challenges to flight control. The National Aeronautics and Space Administration (NASA) developed the "Learn-to-fly "(L2F) algorithm, in the Transforming Aviation Concept Program (TACP), with the purpose to update the development model of aircraft. The traditional aircraft development process includes wind tunnel tests, aero-fluid dynamics calculations (CFD), simulation development, control law design, and the final flight test sequence, and iterates continuously [12][13][14]. Inevitably, during the flight test, the aerodynamic model needs to be continuously updated. Whenever the aerodynamic model needs to be updated, the development process will be iterated once. L2F combines real-time nonlinear aerodynamic modeling with autonomous control law design. The advantages of L2F include the use of aerodynamic models based on real-time identification of flight data in the control law design. Therefore, there is no need to modify the Reynolds number, blockage, boundary layer turbulence, etc [15][16][17]. The control system design is developed based on the actual flight dynamics response rather than the simulation results. As L2F technology matures, new aircraft designs may not require the ground test [18][19][20]. The three pillars of L2F theory are the hot spots of research in recent years: 1) real-time nonlinear aerodynamic modeling; 2) "learning" control law design method; 3) guidance algorithm. In the L2F research, the flight test aerodynamic modeling method is combined with the real-time guidance and learning control law design method on the aircraft and applied in flight. These technologies enable the aircraft to learn how to fly [21][22].
Many scholars besides NASA have also made thorough research on the control of the hypersonic vehicle at widerange Mach numbers. In Refs. [23][24], several original contributions were made by X. W. and Q. Qi, who developed a fuzzy optimal tactic which deals with the tracking control problem of the hypersonic vehicle subject to unknown dynamics. Recently, they made an effort to apply a simplified finite-time fuzzy neural controller with prescribed performance to waverider aircraft which could guarantee satisfied real-time performance with low-complex structure. X. Bu presented a funnel non-affine controller applying neural approximation for prescribed tracking of air-breathing hypersonic vehicles. The desired transient performance and steady-state performance are ensured for both tracking errors [25].
Trajectory Linearization Control (TLC) is a nonlinear control method based on differential-algebraic spectrum theory, which has been applied to the design of X-33's ascent attitude control system in NASA's advanced guidance and control project. It is an advanced control method for hypersonic vehicle control at present [26]. TLC and PID will be performed as comparative groups, reflecting the control effect of the adaptive integration method based on the desired pole assignment.
In this paper, the real-time aerodynamic and thrust data of aircraft are obtained based on the parametric model of the hypersonic vehicle at wide-range Mach numbers. A comprehensive control algorithm based on pole assignment is adopted. On the premise of knowing the real-time aerodynamic parameters of the aircraft, combined with the concept of learning flight, it is hoped that the interaction in flight dynamics, aerodynamics, thermodynamics, frequency and dynamics of the control system of the hypersonic vehicle at wide-range Mach numbers can be handled with a low-complex control structure, which is expected to improve the control quality.
The contributions of this paper mainly include the following four points: The parametric model of hypersonic vehicle at wide-range Mach numbers is constructed and the flight data under various working conditions are calculated.
The adaptive controller design based on the expected pole assignment is completed.
According to the aerodynamic real-time identification results, the state feedback matrix is designed.
The comparative simulation is completed, and the algorithm to solve the aerodynamic/thrust coupling problem is verified. This paper is structured as follows: In section 2, the mathematical model and the essential method of aerodynamic and propulsion characteristics analysis of the hypersonic vehicle at wide-range Mach numbers are expounded. The dynamic analysis of the hypersonic vehicle at wide-range Mach numbers is carried out in section 3. Section 4 describes the control scheme based on pole assignment and the adaptive control gain compensation method. Section 5 gives the simulation comparison results and section 6 gives discussion. a parallelogram or triangle. The aerodynamic forces and moments on the panels are calculated, and the total aerodynamic forces and moments are obtained by adding the calculated results. Identifying a panel need to determine the centroid coordinates, direction, and area. The quadrilateral panel is used for specific explanations.
The coordinates of the four vertices of the quadrilateral panel are 1 2 3 4 , , , Q Q Q Q , and the normal vector outside the panel is calculated by: 12 12 n TT TT Where 12 TT ， are the vector obtained by subtracting the coordinates of two groups of non-adjacent points.
The panel plane can be determined by the normal vector of the panel and the center of the panel. The four vertices are projected to the panel plane, and the points projected on the panel plane are called "corners". The coordinates of the four corners are: Where P is the median coordinate of 4 nodes. The coordinate system is established based on the panel, the origin is in the center of the panel, and the three axial directions are: The transformation from computational coordinate system to panel coordinate system: ,, The coordinates of the four corners in the panel coordinate system are as follows: The coordinates of that centroid of the panel in the panel coordinate system are as follows: The coordinate of that centroid of the panel in the calculated coordinate system is as follow: The area of the panel is: According to the engineering estimation method, the pressure on the ith plane is obtained.
P ∞ is local static pressure and , ci q is local dynamic pressure.
The total aerodynamic force and aerodynamic moment on the aircraft can be obtained by adding the aerodynamic force and aerodynamic moment vectors on all panels.
is the distance vector from the centroid of the panel to the centroid of aircraft, and ,, xi yi zi n n n are the components of n on ,, The panel effect of airfoils is shown in Figure 2. Each small panel is represented by a small arrow pointing to its normal vector. The red part is the elevator, which can deflect with the input angle. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
In thrust estimation, scramjet is divided into the inlet, isolator, combustion chamber, and nozzle.
In the inlet part, shock wave, expansion wave, and their intersection theory are used to describe the flow parameters. All of the shock waves and expansion waves in the flow region produced by the turning of the inlet surface and the intersection of shock waves and expansion waves are calculated. These waves divide the two-dimensional area into several small areas. The internal airflow parameters in each small area remain constant, which are determined by the shock waves or expansion waves in front. If there is a detached shock wave or the calculated wave surface is more than a certain number, the average airflow parameters here will be calculated instead. The airflow parameters from here to the isolation section can be determined according to the frictional variable cross-section pipe flow formula (formula 16). The algorithm flow is shown in Figure 3.   The quasi-one-dimensional model is adopted to calculate the airflow parameters in the isolation section and combustion chamber, and the airflow friction, fuel injection, length change of pre-combustion shock wave train, reaction rate are out of consideration. The basic conservation equations of airflow in the isolation section and combustion chamber are obtained, and the equations need to combine experimental or empirical formulas to get solved. Finally, the airflow parameter distribution law in the isolation section and combustion chamber is [35][36]: This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
In which: The plume method is adopted at the nozzle reflecting the coupling of flight and propulsion [37][38]. The plume model is based on the equal air pressure above and below the shear layer (see Fig 3).

B. DYNAMIC MODEL OF THE HYPERSONIC VEHICLE AT WIDE-RANGE MACH NUMBERS
The force of the hypersonic vehicle is divided into three categories: one: vertical downward gravity, two: thrust in the direction of the body axis, three: aerodynamic force, the aerodynamic force is decomposed into a lift perpendicular to the speed, and the resistance is opposite to the speed direction. Combined with Newton's law, the following equations can be obtained

A. DYNAMIC CHARACTERISTIC VERIFICATION OF PARAMETRIC MODLE
Validating the effectiveness of the developed model is an essential step to ensure the developed model is reliable. In this part, the trim capacity of lift to gravity and thrust to drag is put to the proof. The trim elevator deflection at each angle of attack is also a key concern. The thrust eccentricity, the additional moment generated by the internal flow on the afterbody flow field and the balancing of the external aerodynamic moment are taken into consideration. The static stability analysis under the equilibrium characteristic points is carried out, and the corresponding static stability coefficients and maneuverability at different angles of attack are solved respectively. It is proved that the model is reliable and can be applied to control.

. Model validation chart. (a) Thrust-drag balance test; (b) Liftgravity balance test; (c) Trimming rudder deflection at different angles of attack and Mach number; (d) Static stability test; (e) Elevator efficiency test
Form Figs. 5(a-e), we can conclude that: 1) the drag reaches the minimum near the angle of attack of 0 degrees, and the fuel equivalent ratio has no prominent influence on the drag. The thrust increases with the increase of angle of attack or fuel equivalence ratio, which shows that the developed model has trim capability in a large angle of attack range. 2) The lift can be balanced with the maximum gravity near the angle of attack of three degrees.
3) The trim ability of the elevator decreases with the increase of Mach number, but the pitching moment can be trimmed within a reasonable angle of attack.
4) The developed model between Ma4-6 is statically unstable, and the maximum static instability is near 3 degrees angle of attack. 5) The elevator efficiency of the developed model is within a reasonable range.

B. FLIGHT MECHANICS ANALYSIS
Undisturbed motion is space unsteady flight, and the missile motion equation is linearized with small disturbance: To make the above formula easy to write, the simplified symbol of the equation coefficient is introduced.  a can represent the static stability of aircraft. If 24 0 a  , the aircraft is statically stable, otherwise it is statically unstable. 25 a can represent the maneuverability of the aircraft, that is, the moment change that can be produced when the elevator deflects by one unit.
The free disturbance motion of the hypersonic vehicle at wide-range Mach numbers is analyzed, and the free disturbance motion equations are obtained by The following special solutions in the form of exponential functions of the equations are: After eliminating the common term, the dynamic stability of aircraft can be judged by solving the characteristic equation.
Ignore the velocity which changes in a long period. The transfer function matrix in a short period is given.

IV. ADAPTIVE CONTROL BASED ON POLE ASSIGNMENT
According to the longitudinal short-period disturbance Where: 22 can form a matrix to make the system asymptotically stable, and if The eigenvalue of A + BK is the pole of the regulator. If the pole of the regulator is located in the left half-plane of the x .
The problem of arranging the closed-loop poles to the desired position is called the pole arrangement. What should be noted is it's possible to modify any pole arrangement of the system only if the given system is entirely controllable.
There are four algorithms commonly used in pole placement of MIMO high-order systems. 1) Turn the multiinput system into a single-input system; 2) Convert the state space expression into the Laenberger controllable canonical form, and then the state feedback gain K can be solved; 3) Solve the T matrix of the Sylvester equation, if the T matrix is nonsingular, the K is obtained. 4) The state feedback gain K is obtained by the linear-quadratic optimal control method.
Comparing the four algorithms, the disadvantage of the first algorithm is that the single-input system generated by multi-input systemization is not unique, and the elements of the feedback gain matrix i k calculated by this algorithm are often too large. The disadvantage of the second algorithm is that the calculation of the transformation matrix which can control the canonical form is tedious. The third algorithm cannot guarantee that the T matrix is nonsingular. The fourth algorithm is subjective in the balance between error and energy consumption, that is, the setting of the Q matrix and the R matrix. In this paper, the state feedback matrix is obtained by taking the expected poles, and the logic of the control matrix is: 11 12 13 14 It can be seen that in the process of calculating elevator deflection components  is an irrelevant term. The fourth-order system is divided into a third-order system and a first-order system according to the logic of controlling attitude by elevator and controlling thrust by fuel equivalence ratio. The third-order system corresponds to the angle of attack control and has two dominant poles and one non-dominant pole. The dynamic index of the system step response is mainly determined by two dominant poles.
The damping ratio and natural frequency are calculated according to the performance index of the second-order system, and the dominant pole is selected using the formula (32). The relationship between adjustment time and overshoot and damping ratio and natural frequency is shown in formula (33): Where  is called the damping ratio of the system; and n  , , the undamped natural frequency. It is customary for the third pole to take a value more than 5 times the real part of the dominant pole. Considering the physical meaning of the state feedback matrix and the requirements of the system dynamic index, the position of the expected pole can be selected: The physical meaning of each element in matrix K is interpreted in detail. It is unreasonable to simply use the mathematical method to solve the matrix K because only four expected poles are used to establish the equation, while there are eight unknown elements, which leads to the non-unique solution of the matrix. The initial K matrix is a suitable value solved by combining the physical meaning of each element of the K matrix with engineering methods and Re[λ(A+BK)]<0 can be guaranteed at this time. Under the influence of uncertainty in flight, some elements of matrix A may get changed, the corresponding elements of matrix K change as compensation, thus ensuring that the value of each element of A+BK remains unchanged. Re[λ(A+BK)]<0 can be always held in this method. The iterative process is as Figure 6. Based on the above dynamic model and controller, the simulation is carried out. The dynamic parameters of the aircraft are shown in Table Ⅱ:   TABLE Ⅱ  THE ONTOLOGICAL PARAMETERS OF THE HYPERSONIC VEHICLE AT WIDE-RANGE MACH NUMBERS According to the previous section, the desired pole is selected as: According to the formula (32)(33), it is expected that the angle of attack setting time corresponding to this set of poles will be about 3.2 seconds, and the maximum overshoot will be about 20%.
Under the desired poles selected above and in combination with the requirements of the system dynamic response index, the tuning state feedback matrix is solved as . The flight process of the aircraft is set at an altitude of 25,000 meters and a speed of Mach 6. The initial angle of attack is 0 degrees and adjusted to 1 degree. The initial fuel equivalence ratio is 0, which is adjusted to 0.85 and simulated in the interval of 0 to 10 seconds. The simulation results include the dynamic response process under normal conditions, dynamic response process of the elevator after adding 20% uncertainty, and dynamic response process of controller adaptive compensation while adding 20% uncertainty to the elevator. PID and trajectory linearization control (TLC) are simulated as comparative methods. The simulation results are shown in figure 8: Description Value

Ⅵ. DISCUSSIONS AND CONCLUSIONS
From Figure 7, due to the increase of the order of transfer function, the Bode diagram of the hypersonic vehicle at wide-range Mach numbers control system is more complex with the coupling of flight and propulsion. The amplitude of high-frequency band in (a) and (d) is small, which shows that the impaction from an executive body to the controlled objects (i.e., from the elevator to the angle of attack and from the fuel equivalence ratio command to the fuel equivalence ratio) has a strong ability to suppress noise. However, the high-frequency amplitude of the coupling control process as shown in (b) and (c) is large, which shows that they have a weak ability to suppress high-frequency noise. On the contrary, their low-frequency amplitude is higher, which shows that their steady-state is more accurate.
From Figure 8(a-b), the maximum overshoot of the PID control system for the angle of attack is 43.4%, and the setting time is 3.36 seconds Differ from PID, which linearizes the nonlinear system in only one working condition, LTC linearization in many different working conditions can reduce the overshoot of maximum angular velocity and angle of attack, which shows that it has certain advantages compared with PID. The maximum overshoot reduced to 27 LTC. Adaptive control based on desired pole assignment can adjust the controller according to the variation of the realtime flight parameter, which decrease the overshoot of angle of attack to 20.2%. The setting time is 3.16 seconds, which meets the requirements of the desired pole for the dynamic response index of the system. Results of simulation show that the adaptive controller can obviously improve the control quality of the system with uncertainties. From(c-e), although the adjustment time of the three cases is the same, it can be seen that the adaptive control can reduce the maximum peak value of the curve. It can be seen from c and d that the response curve of adaptive control is closer to the steadystate value as a whole.
In this paper, an adaptive controller based on pole assignment is adopted to suppress the disturbance caused by uncertainty in flight. Compared with the widely used methods such as trajectory linearization, the adaptive controller based on pole assignment can adjust the control parameters according to the real-time flight data identification, thus achieving a better integrated-control effect.
The hypersonic vehicle at wide-range Mach numbers model is built mainly according to the parametric method. On this basis, the detailed aerodynamic data and thrust data of the aircraft under various working conditions are calculated according to the panel method and the others, which can show a large proportion of the uncertainties and time variability in the large flight envelope. The flightpropulsion coupling characteristics of the hypersonic vehicle at wide-range Mach numbers are described by using the state-space equation and transfer function. On the basis of the simplified state-space equation, this paper gives an integrated control strategy for the hypersonic vehicle at wide-range Mach numbers by using the pole assignment method. Realtime identification based on flight data adjusts the controller parameters to compensate for the uncertain disturbance in flight while keeping the desired pole unchanged. The simulation results show that the pole assignment method can be used in the integrated control of the hypersonic vehicle at wide-range Mach numbers, and the influence of uncertain disturbance on the system during flight can be compensated by using the identification method. It shows that the adaptive controller based on the pole assignment is an effective in dealing with the coupling effect of flight and propulsion of the hypersonic vehicle at wide-range Mach numbers and the uncertainty in large flight envelope with low-complexity control framework.
Further research includes the following points: (1) In this paper, the control of fuel equivalence ratio is simplified, and the dynamic process related to flight attitude is ignored. In the future, the dynamic process of fuel equivalence ratio can be analyzed and studied in more detail according to the parametric model; (2) In terms of application objects, parametric models can be used for dynamic analysis of various aircraft. The adaptive control method based on pole assignment can be applied to the coupling control of MIMO systems in different fields; (3) Empirical coupling control values are adopted in setting control coupling items, and identification of coupling can be added in the future to adjust coupling item parameters in real-time.